How can I have multiple separate texture maps for separate objects? Is it possible?
Yes, it's possible. If you have the wings and ears in a separate object, just create a new material, and select there the image texture you want.
The relationship is the following: Object → Mesh → Material → UVMap → Image Texture.

A mesh can have more than 1 material. In the Edit mode you can select faces and assign them to a material in the material tab.

In the Image Editor you can select the texture images to inspect them or paint on them.

In the Texture Editor, the UV map and the texture are specified in the Texture Slot settings. Blender can read this pair from the material or you can specify the there. This defines on which image and where the pixels go when you paint on the 3D model, and vice versa.

In the Shader Editor you can define with nodes the UV Map and the Image Texture that a material uses.

And in the Object Data Properties &gt; UV maps you can select the active (default) UV map for the viewport and for the render.